of 1 36
Life A Mathematical Construct
By
Ian Beardsley
Copyright © 2021 by Ian Beardsley
of 2 36
Table of Contents
Abstract…………………………………………..3
Silicon and Carbon……………………………….4
Introduction……………………………………….16
Molecular Geometry………………………………17
Bone As A Mathematical Construct………………18
Breaking Down Bone……………………………..26
Comparing The Elements Within A Compound…..31
Conclusion………………………………………..35
of 3 36
Abstract
Here an attempt is made to show that biological life (with biological life core elements H, N, C, O) and
artificial intelligence life (with core elements P, B, Si, Ge) are connected to one another if we can show
they are not just chemical constructs, but mathematical constructs, which is done by showing there is a
connection between Si and C.
of 4 36
Silicon and Carbon
We guess that artificial intelligence (AI) has the golden ratio, or its conjugate in its means geometric,
harmonic, and arithmetic by molar mass by taking these means between doping agents phosphorus (P)
and boron (B) divided by semiconductor material silicon (Si) :
Which can be written
We see that the biological elements, H, N, C, O compared to the AI elements P, B, Si is the golden ratio
conjugate (phi) as well:
So we can now establish the connection between artificial intelligence and biological life:
Which can be written:
Where HNCO is isocyanic acid, the most basic organic compound. We write in the arithmetic mean:
Which is nice because we can write in the second first generation semiconductor as well (germanium) and
the doping agents gallium (Ga) and arsenic (As):
PB
Si
=
(30.97)(10.81)
28.09
= 0.65
2PB
P + B
1
Si
=
2(30.97)(10.81)
30.97 + 10.81
1
28.09
= 0.57
0.65 + 0.57
2
= 0.61 ≈ ϕ
PB(P + B) + 2PB
2(P + B)Si
≈ ϕ
C + N + O + H
P + B + Si
≈ ϕ
(P + B + Si )
PB(P + B) + 2PB
2(P + B)Si
≈ (C + N + O + H )
PB
[
P
Si
+
B
Si
+ 1
]
+
2PB
P + B
[
P
Si
+
B
Si
+ 1
]
≈ 2HCNO
[
PB +
2PB
P + B
+
P + B
2
][
P
Si
+
B
Si
+ 1
]
≈ 3H NCO
of 5 36
Where
Where ZnSe is zinc selenide, an intrinsic semiconductor used in AI, meaning it doesn’t require doping
agents. We now have:
Germanium And Carbon
We could begin with semiconductor germanium (Ge) and doping agents gallium (Ga) and Phosphorus (P)
and we get a similar equation:
,
In grams per mole. Then we compare these molar masses to the molar masses of the semiconductor
material Ge:
Then, take the arithmetic mean between these:
We then notice this is about the golden ratio conjugate, , which is the inverse of the golden ratio, .
. Thus, we have
1.
2.
[
PB +
2PB
P + B
+
P + B
2
][
P
Si
+
B
Si
+ 1
]
≈ H NCO
[
G a
G e
+
As
G e
+ 1
]
Z n
Se
≈
[
P
Si
+
B
Si
+ 1
]
[
Ga
Ge
+
As
Ge
+ 1
]
PB
(
Z n
Se
)
+
2PB
P + B
(
Z n
Se
)
+
P + B
2
(
Z n
Se
)
≈ H NCO
2G a P
G a + P
= 42.866
G a P = 46.46749
2G a P
G a + P
1
G e
=
42.866
72.61
= 0.59
G a P
1
G e
=
46.46749
72.61
= 0.64
0.59 + 0.64
2
= 0.615
ϕ
Φ
ϕ ≈
1
Φ
G a P(G a + P ) + 2G a P
2(G a + P )G e
≈ ϕ
G a P(G a + P ) + 2G a P
2(G a + P )Si
≈ Φ
of 6 36
This is considering the elements of artificial intelligence (AI) Ga, P, Ge, Si. Since we want to find the
connection of artificial intelligence to biological life, we compare these to the biological elements most
abundant by mass carbon (C), hydrogen (H), nitrogen (N), oxygen (O), phosphorus (P), sulfur (S). We
write these CHNOPS (C+H+N+O+P+S) and find:
A similar thing can be done with germanium, Ge, and gallium, Ga, and arsenic, As, this time using
CHNOPS the most abundant biological elements by mass:
We can also make a construct for silicon doped with gallium and phosphorus:
And for germanium doped with gallium and phosphorus:
CH N OPS
G a + A s + G e
≈
1
2
[
G a As +
2G a As
G a + A s
+
G a + A s
2
][
G a
G e
+
As
G e
+ 1
]
≈ CH NOPS
[
G a
Si
+
As
Si
+ 1
]
G a As
(
O
S
)
+
2G a As
G a + A s
(
O
S
)
+
G a + A s
2
(
O
S
)
≈ CH NOPS
O
S
≈
[
Ga
Ge
+
As
Ge
+ 1
]
[
Ga
Si
+
As
Si
+ 1
]
G a As(G a + As) + 2G a A s
2(G a + A s)G e
≈ 1
C + H + N + O + P + S
G a + A s + G e
≈
1
2
(C + N + O + H ) ≈
2(G a + P )Si
G a P(G a + P ) + 2G a P
(P + B + Si )
HNCO ≈
2(G a + P )Si
(G a + P )
[
G a P +
2GaP
Ga + P
]
(P + B + Si )
HNCO ≈
2(P + B + Si )Si
G a P +
2GaP
Ga + P
G a P(G a + P ) + 2G a P
2(G a + P )G e
≈ ϕ
of 7 36
Here is a table of the AI biological equations…
[
G a P +
2G a P
G a + P
+
G a + P
2
][
P
G e
+
B
G e
+
Si
G e
]
≈ H NCO
[
G a
G e
+
As
G e
+ 1
]
G a P
(
B
S
)
+
2G a P
G a + P
(
B
S
)
+
G a + P
2
(
B
S
)
≈ H NCO
of 8 36
The Fundamental AI Bioequations
[
PB +
2PB
P + B
+
P + B
2
][
P
Si
+
B
Si
+ 1
]
≈ H NCO
[
G a
G e
+
As
G e
+ 1
]
[
G a As +
2G a As
G a + A s
+
G a + A s
2
][
G a
G e
+
As
G e
+ 1
]
≈ CH NOPS
[
G a
Si
+
As
Si
+ 1
]
[
G a P +
2G a P
G a + P
+
G a + P
2
][
P
G e
+
B
G e
+
Si
G e
]
≈ H NCO
[
G a
G e
+
As
G e
+ 1
]
HNCO ≈
2(P + B + Si )Si
G a P +
2GaP
Ga + P
PB(P + B ) + 2PB
2(P + B)Si
≈ ϕ
G a As(G a + As) + 2G a A s
2(G a + A s)G e
≈ 1
G a P(G a + P ) + 2G a P
2(G a + P )G e
≈ ϕ
G a P(G a + P ) + 2G a P
2(G a + P )Si
≈ Φ
C + N + O + H
P + B + Si
≈ ϕ
C + H + N + O + P + S
G a + A s + G e
≈
1
2
Z n
Se
≈
[
P
Si
+
B
Si
+ 1
]
[
Ga
Ge
+
As
Ge
+ 1
]
O
S
≈
[
Ga
Ge
+
As
Ge
+ 1
]
[
Ga
Si
+
As
Si
+ 1
]
of 9 36
Also, to look at life as a mathematical construct, look at CH4 which is one of the primordial earth
substances (methane) and is part of making amino acids, the building blocks of life. Since H is H1+ and C
is C4-, then to have neutral methane:
To look at this mathematically (as a mathematical construct) think of it in terms of geometry: Since C is
C4- meaning it has four valence electrons, think of it as a regular fours sided shape (a square) and H as a
regular octagon or truncated square:
If each side of the carbon square is one, then the radius r1 is square root 2 over 2:
of 10 36
Since we are saying hydrogen is a regular octagon then its r, r2 is the quantity one plus square root of 2
over 2:
of 11 36
Which takes us to the square root of three. Let us consider the other primordial substance that was in the
early earth that gave rise to amino acids, the building blocks of life, ammonia (NH3). Since N (nitrogen)
is N3- and hydrogen (H) is H1+, we have:
If the side is s=1, then the ratio of the side s to the radius r3 is:
Is our square root of three from considering methane (CH4). Thus, where carbon (C4-) represents (is
represented by) a square, then ammonia (NH3) has nitrogen (N3-) represented by an equilateral triangle.
The other pivotal substance that was in the early earth used to make amino acids, the building blocks of
life, was water (H2O). Since H is H1+ and O is O2- we have:
1
1
3
=
s
r
3
= 3
of 12 36
I have formed equations for the terrestrial planets, Venus and Mars, in other works both in terms of
artificial intelligence core elements Si and Ge, and mathematical constants Euler’s number e and the
golden ratio conjugate phi.
Not only is the Venus equation perfect in the equation set in terms of AI elements, but in the Mars
equation in terms of and in the equation set in terms of these constants. We have:
Let us solve for x in:
1.52=0.72x
x=2.11111=19/9
This is very close to fluorine (F) over beryllium (Be):
Beryllium (see illustrations) is pivotal to the production of carbon, which is in turn the core element to
biological life compounds. Stars produce carbon by combining two helium (He) atoms to make
Beryllium. The Beryllium then combines with another helium atom to make carbon. Beryllium rarely
occurs in Nature because it is usually depleted in the reaction in stars using it to make heavier elements. I
found a connection of beryllium to carbon by way of silicon. The radius of silicon is Si=0.118nm. If we
say it is inscribed in a dodecagon (12 sided) regular polygon of side 1, then carbon inscribes in a regular
octagon (8 sided) of side 1. This is the eight of The beryllium-8 (four protons, four neutrons) that makes
carbon by combining with helium.
And here we show that silicon (core to artificial life) is connected to (C) biological life as a mathematical
construct. We find if we inscribe carbon in a regular octagon radius C=0.077nm then silicon Si=0.118 nm
is inscribed in a regular dodecagon (12 sided polygon) if they are to connect to one another with sides of
the same length.
ϕ
e
venu s =
1
G e
2
2 SiG e +
Si
3
Ge
1 +
Si
2
Ge
2
= 0.72AU
Ma rs = ϕ
2
e
2−ϕ
= 1.52A U
F
Be
=
19.00g /m ol
9.01g /m ol
= 2.108768 ≈ 2.1
of 13 36
of 14 36
of 15 36
Flourine is the most electronegative element making it highly reactive. It reacts with everything but
argon, neon, and helium. Flourine combines with carbon to make fluorocarbons. The primary mineral
source is fluorite and is mined for steelmaking which turn results in byproducts used aluminum refining.
Its electron configuration is , giving it seven outer electrons so that it needs one more to be
filled, to give it eight outer electrons because it tends to capture an electron to make it isoelectronic with
the noble, or inert gas neon.
Thus we have the following Equation that through the planets Venus and Mars, the solid planets directly
on either side of the Earth, relates artificial intelligence to biological life:
We have that at the beginning of the Universe hydrogen and helium were created. Then the stars formed
and synthesized these into the heavier elements. I find if we include in the category of life not just the
biological elements, but the AI elements, we can find a mathematical equation for a pattern in the periodic
table of the elements that predicts the synthesis of such elements in stars. For instance, Beryllium 8 plus
helium 4 synthesizes to make the biological core element carbon C. Magnesium plus helium 4
synthesized to make the core AI element silicon Si. If we say that Element 4 is Beryllium Be and write it
, and helium He is element 2 and write it , and use this convention for all of the elements, we have
for the production of these elements by stars, and their molar masses in the periodic table the following
equation:
1S
2
2 S
2
P
5
1
G e
2
2 SiG e +
Si
3
Ge
1 +
Si
2
Ge
2
F
Be
= ϕ
2
e
(2 −ϕ)
E
4
E
2
E
2n−2
+ E
2
= E
2i−2
= (4k + 4)g /m ol
n = (3,4, 5,6, …)
i = (4,5, 6,7, …)
k = (2,3, 4,5, …)
of 16 36
Introduction
In my works The Mathematical Nature of Life (Beardsley 2021) and Perfect Equations Beardsley (2021) I
set out to find if the the elements and compounds characteristic of life and artificial intelligence (AI) do
not just conform to chemical law, but if they are purely mathematical independently of the use of
chemistry to describe them, and if they are connected to one another. The simplest example of this for
biological life and AI would be that the most basic organic compound is HNCO (isocyanic acid) where H
(hydrogen), N (nitrogen), C (carbon), and O (oxygen) are the most abundant biological elements. Indeed
biological elements are for the most part organic, which means they are made of long chains using carbon
with hydrogen, which they can form because C is C4- and H is H+ meaning we can have:
And in isocynaic acid we have:
H-N=C=O
Where H is H+, N is N3-. C is C4-, O is O2-, the H uses its single bond with one from nitrogen, leaving
N2- or two bonds which go to C leaving for it C2- which goes to oxygen that needs it because it is O2-.
Thus all is satisfied by chemical law. In my search for mathematical law, I find it exists in the case of
HNCO and the AI semiconducting element silicon (Si) and its doping agents P and B as such (by molar
mass):
This paper strives to break down such mathematical equations for biological life and artificial intelligence
into their components to find what is acting to create such constructs. In the second book I actually
brought the planets into the mix with some very interesting results. As another example, water and air, the
main physical constituents that interact with life we have:
C + N + O + H
P + B + Si
≈ ϕ
ϕ =
a
b
=
5 − 1
2
c = b + a
a
b
=
b
c
H
2
O
air
≈ ϕ
air = 0.25O
2
+ 0.75N
2
of 17 36
By molar mass for air as a mixture (not a compound). With this air is 29.0 grams per mole.
Molecular Geometry
We will want to break down our equations into the components of their geometric relationships and see if
they predict the bond angles of some of the basic substances considered. We will look here at linear,
trigonal planar, and tetrahedral.
Linear, like CO2 (carbon dioxide) its bond angle is 180 degrees:
Trigonal planar, like SO3 (sulfur trioxide) its bond angle is 120 degrees:
O
|
S
/. \
O. O
That is, S is at the center and the O atoms are 120 degrees apart due to the even division of 360/120=3.
Tetrahedral, like methane (CH4) one of the the primordial gases that may have contributed to making
some of the amino acids, the building blocks of life as show by Miller and Urey in the early origins of
life:
This is 109.5 degrees apart from arcos (1/3) = 109.5
But what if we are considering not just neutral molecules but polyatomic anions that have a net charge. In
such instances, the free electron pairs compress the expected 120 degree bond angle in the atoms around
the central atom to 115 degrees as with the nitrite ion NO2-:
of 18 36
Similarily we have for O3 (ozone) that the bond angle is 116 degrees in its deviation from 120 degrees.
The configuration is:
Both of these anions are important to the life and the theory of how life forms. O-zone is more of a
physical component in that in the stratosphere it absorbs UV radiation harmful to life.
Bone As A Mathematical Construct
What better place to begin than with than bone as it is the basic framework around which skeletal life is
structured, the vertebrates. Here is what I found in bone as a mathematical construct:
In my exploration of the connection between biological life and AI the most dynamic component is that of
bone. It affords us the opportunity to look at:
Multiplying Binomials
Completing The Square
The Quadratic Formula
Ratios
Proportions
The Golden Ratio
The Square Root of Two
The Harmonic Mean
of 19 36
Density of silicon is Si=2.33 grams per cubic centimeter.
Density of germanium is Ge=5.323 grams per cubic centimeter.
Density of hydroxyapatite is HA=3.00 grams per cubic centimeter.
This is
where
Where HA is the mineral component of bone, Si is an AI semiconductor material and Ge is an AI
semiconductor material. This means
The harmonic mean between Si and Ge is HA,…
This is the sextic,…
Which has a solution
Where x=Si, and y=Ge. It works for density and molar mass. It can be solved with the online Wolfram
Alpha computational engine. But,…
3
4
Si +
1
4
G e ≈ H A
H A = Ca
5
(PO
4
)
3
OH
Si
H A
Si +
[
1 −
Si
H A
]
G e = H A
2 SiG e
Si + G e
≈ H A
x
2
(x + y)
4
− x y(x + y)
4
+ 2x y
2
(x + y)
3
− 4x
2
y
2
(x + y)
2
= 0
Si
G e
=
1
2 + 1
1
H A
2
Si
2
−
G e
H A
2
Si +
[
G e
H A
− 1
]
= 0
Si =
1
2
G e
±
H A
G e
H A
2
−
4G e
H A
+ 4
Si = G e − H A
of 20 36
Si
H A
Si +
[
1 −
Si
H A
]
G e = H A
Si
2
H A
+ G e −
Si
H A
G e ≈ H A
1
H A
Si
2
−
G e
H A
Si + G e ≈ H A
1
H A
2
Si
2
−
G e
H A
2
Si +
G e
H A
≈ 1
1
H A
2
Si
2
−
G e
H A
2
Si +
G e
H A
− 1 ≈ 0
1
H A
2
Si
2
−
G e
H A
2
Si +
[
G e
H A
− 1
]
= 0
of 21 36
We see that the square of the binomial is a quadratic where the third term is the square of one half the
middle coefficient. This gives us a method to solve quadratics called completing the square:
(x + a)(x + a) = x
2
+ 2a x + a
2
(x + a)
2
= x
2
+ 2a x + a
2
a x
2
+ bx + c = 0
a x
2
+ bx = − c
x
2
+
b
a
x = −
c
a
(
1
2
b
a
)
2
=
1
4
b
2
a
2
x
2
+
b
a
x +
1
4
b
2
a
2
= −
c
a
+
1
4
b
2
a
2
(
x +
1
2
b
a
)
2
=
b
2
− 4a c
4a
2
x +
b
2a
=
±
b
2
− 4a c
2a
x =
−b
±
b
2
− 4a c
2a
of 22 36
1
H A
2
Si
2
−
G e
H A
2
Si +
[
G e
H A
− 1
]
= 0
x =
−b
±
b
2
− 4a c
2a
a =
a
H A
2
b = −
G e
H A
2
c =
[
G e
H A
− 1
]
b
2
− 4a c =
G e
2
H A
4
− 4
1
H A
2
[
G e
H A
− 1
]
=
G e
2
H A
4
−
4G e
H A
3
+
4
H A
2
=
1
H A
2
[
G e
2
H A
2
−
4G e
H A
+ 4
]
b
2
− 4a c =
1
H A
(
G e
H A
− 2
)
2
x =
Ge
HA
2
±
1
HA
[
Ge
HA
− 2
]
2
HA
2
=
1
2
G e
±
1
2
H A
[
G e
H A
− 2
]
=
1
2
G e
±
1
2
G e − H A
Si =
1
2
G e +
1
2
G e − H A
Si = G e − H A
of 23 36
Si ≈ Ge − H A
H A ≈
2 SiG e
Si + G e
Si ≈ Ge −
2 SiG e
Si + G e
(Si + G e)G e
Si + G e
−
(Si + G e)Si
Si + G e
−
2 SiG e
Si + G e
= 0
G e
2
− 2SiG e − Si
2
Si + G e
= 0
x
2
− 2x y − y
2
= 0
x
2
− 2x y = y
2
x
2
− 2x y + y
2
= 2y
2
(x − y)
2
= 2y
2
x − y =
±
2y
x = y + 2y
x = y(1 + 2)
x
y
= 1 + 2
y
x
=
1
2 + 1
Si
G e
≈
1
2 + 1
of 24 36
A ratio is and a proportion is which means a is to b as b is to c.
The Golden Ratio
and.
or
a
b
a
b
=
b
c
(
Φ
)
a
b
=
b
c
a = b + c
a c = b
2
c =
b
2
a
a = b +
b
2
a
b
2
a
− a + b = 0
b
2
a
2
− 1 +
b
a
= 0
(
b
a
)
2
+
b
a
− 1 = 0
(
b
a
)
2
+
b
a
+
1
4
= 1 +
1
4
(
b
a
+
1
2
)
2
=
5
4
b
a
= −
1
2
±
5
2
b
a
=
5 − 1
2
a
b
=
5 + 1
2
ϕ =
5 − 1
2
Φ =
5 + 1
2
ϕ =
1
Φ
of 25 36
The mineral component of bone hydroxyapatite (HA) is
The organic component of bone is collagen which is
We have
%
Ca
5
(PO
4
)
3
OH = 502.32
g
m ol
C
57
H
91
N
19
O
16
= 1298.67
g
m ol
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
= 0.386795722
ϕ = 0.618033989
1 − ϕ = 0.381966011
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
≈ (1 − ϕ)
0.381966011
0.386795722
100 = 98.75
Si
G e
=
28.09
72.61
= 0.386861314 ≈ (1 − ϕ)
Si
G e
≈
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
of 26 36
Breaking Down Bone
Essentially, in our mathematical formulation of bone, we had that
Which resulted in that the AI elements:
By way of the mineral component of bone HA (hydroxyapatite) is the harmonic mean between Silicon
and Germanium the primary semiconductor elements, which are really the skeleton on AI. Thus, we need
to break down the harmonic mean between Si and Ge into its geometric representation, and through find
what its components are if we are to get any sense of the dynamics. Here I do that in the following
illustration:
Si ≈ Ge − H A
H A ≈
2 SiG e
Si + G e
Si
G e
≈
1
2 + 1
of 27 36
of 28 36
We see that through bone Si and Ge predict an angle of about 116 degrees. This is not the case of linear at
180 degrees, or tetrahedral pyramidal at 109.5 degrees, but is the instance of trigonal planar, but not of
neutral molecules, which is 120 degrees, but of trigonal planar for polyatomic anions such as the nitrite
ion:
And O-zone (Not an anion but has free electrons due to a single bond):
We also will want to look at that aspect of the other mathematical constructions we found for bone:
Which means:
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
≈ (1 − ϕ)
Si
G e
=
28.09
72.61
= 0.386861314 ≈ (1 − ϕ)
Si
G e
≈
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
Si
G e
=
1
2 + 1
2 − 1 ≈ 1 − ϕ
of 29 36
Which means we want to explore this as well:
of 30 36
The thing that is important here is that both and being irratiational, are the operative components to
And this is where we come to some sort of a conclusion as to how square root of two and the golden ratio
in molar mass and density of bone might determine something physical about Nature. We consider that
such physical proportions in the lattice that makes bone, carry through to the proportions found in
humans:
Where we find the square root of two is interesting; we are all familiar with the golden ratio phi in the
human body. For instance in the height divided by the distance from the bottom of the feet to the navel.
But, the answer comes from archaeology.
The intermembral index compares the forelimbs of vertebrates to their hindlimbs. A ratio greater than one
means the forelimbs are longer than the hindlimbs and less than one the hindlimbs are longer. It is this
ratio that tells paleontologists a great deal about the manner of propulsion of a vertebrate.
The chimpanzee index is 106, or 1.06 in other words as a fraction, meaning their forelimbs are longer
than their hindlimbs compared to humans, which are around 68-70 or 0.68 to 0.7 meaning their hindlimbs
are longer than their forelimbs. Thus we see they have their forelimbs are longer for climbing, arm
hanging and swinging activities. The longer hindlimb of humans means they depend sole on these for
propulsion in bipedal walking. Lucy, the 3.2 million year old hominid (Australopithecus Afarensis) has
index 88 (0.88) intermediate between humans and chimpanzees, and this due to a shortened humerus, not
elongated thigh, showing arm length reduced first in the evolutionary trend toward being bipedal. She
probably used hindlimb for bipedal propulsion and forelimbs for climbing.
Measuring myself I find I have humerus+radius=22”, and femur+tibia=32”. My intermembral index is
about i=22/32=0.6875. And here is our :
Are we evolving towards an intermembral index of ?!
ϕ
2
2 − 1 ≈ 1 − ϕ
2
1
i
=
1
0.7
= 1.42857 ≈ 2 = 1.414
2
of 31 36
Comparing the Elements Within a Compound
Here instead of comparing one compound to another, we compare the elements in the compound to one
another and the compound itself. We start with bone, for which as we have said before, the mineral
component is hydroxyapatite (HA) and the organic component is collagen. Thus we have:
Calcium (Ca)=40.078 g/mol
Phospate anion (PO4)=30.973 g/mol
Ca5=200.39 g/mol
(PO)3=92.919g/mol
Hydroxide ion (OH)=17.01g/mol
C=12.01 g/mol
H=1.01 g/mol
N=14.01g/mol
O=16.00 g/mol
We begin comparing:
….
H A = Ca
5
(PO
4
)
3
OH
1.55g /cm
3
white = 1.823g /c m
3
red = 2.2 − 2.34g /c m
3
vi olet = 2.36g /cm
3
bl ack = 2.69g /c m
3
Coll age n = C
57
H
91
N
19
O
16
Ca
5
Ca
5
(PO
4
)
3
OH
=
200.39
502.32
= 0.3989 ≈ 0.4
(PO
4
)
3
Ca
5
(PO
4
)
3
OH
=
92.919
502.32
= 0.18497969 ≈ 0.2
OH
Ca
5
(PO
4
)
3
OH
=
17.01
502.32
= 0.00199 ≈ 0.002
of 32 36
Calcium Ca and carbon C are the primary mineral components of the respective compounds HA and
Collagen. Thus, we call them the i vector. Phosphate and nitrogen are the primary salt components, thus
we call them the j vector. OH and O are the oxygen components, therefore we call them the k vectors. We
have:
Is the angle between the HA and Collagen vectors. Taking the cross product we have:
Which has a magnitude 0.0906. We already know that AB=0.25678 and since:
We have that:
Which is a theta that is around 21 degrees, which we see is right because we got the same value for the
dot product. But just what are we doing when we cross g/mol with g/mol? We look at that:
C
57
C
57
H
91
N
19
O
16
=
684.57
1302.5
= 0.52558 ≈ 0.5
H
91
C
57
H
91
N
19
O
16
=
91.91
1302.5
= 0.07
N
19
C
57
H
91
N
19
O
16
=
266.19
1302.5
= 0.204 ≈ 0.2
O
16
C
57
H
91
N
19
O
16
=
256
1302.5
= 0.1965 ≈ 0.2
A ⋅
B = (Ca
5
)(C
57
) + (PO
4
)
3
N
19
+ (OH )(O
16
) = (0.4)(0.5) + (0.2)(0.2) + (0.002)(0.2) = 0.24
A B = 0.4
2
+ 0.2
2
+ 0.002
2
0.5
2
+ 0.2
2
+ 0.2
2
= 0.25678
A ⋅
B = A Bcosθ
θ = 20.83
∘
i j k
Ca
5
(PO
4
)
3
OH
C
57
N
19
O
16
= 0.0396i − 0.079j − 0.02k
A × B = A Bsi n θ
0.0906 = 0.25678sin θ
si n θ =
0.0906
0.25678
θ = 20.66
∘
of 33 36
It is grams squared per moles squared acting on grams per mole through an angle, which would be like
torque in our instance of grams per mole of the mineral component on the organic matrix through which
it is distributed.
The mineral component of bone is the solid, hard atoms of bone which are embedded in the organic
collagen matrix. So what we are doing is analyzing the forces over which the HA does not tear apart the
collagen. The organic collagen matrix through which the the HA is distributed. So, at this point we might
want to look at the density of bone and the density of collagen to find if the mathematical structure we
have found predicts the experimental strength of bone. If we look at the HA and collagen in such a way
that we get cancellation of grams and end up with moles and cubic centimeters we will be looking at:
But really the moles squared per grams squared is a normalized molar mass, the molar mass in terms of
the molar masses of the compounds of which they are a part. Really we are dealing with unitless numbers.
The density of HA is:
For dry collagen it ranges from 0.0076 and 0.0311 with pore sizes between 250 and 350 a
range in which it scaffolds for functional soft tissue growth. Since the upper value is for strongest bone,
we will use it. We have:
We now divide this by our A dot B which is in grams squared per moles squared:
Now we take the cube root:
We now find if we take the arc sine or arc cosine of this we have one equals the other is 45 degrees.
A ×
B = (g . m ol )(g /m ol )
i + (g /m ol )(g /m ol )
j + (g /m ol(g /m ol )
k
A ×
B = (g
2
/m ol
2
)
i + (g
2
/m ol
2
)
j + (g
2
/m ol
2
)
k
A ×
B = (g
2
/m ol
2
)
2
+ (g
2
/m ol
2
)
2
+ (g
2
/m ol
2
)
2
= g
2
/m ol
2
τ
m ol
2
g
2
⋅
g
cm
3
=
m oles
2
gr a m s ⋅ c m
3
H A = 3.00g /c m
3
g
cm
−3
μm
m ol
2
0.0906g
2
0.0076g
cm
3
= 0.084
m ol
2
g ⋅ c m
3
0.084
m ol
2
g ⋅ c m
3
m ol
2
0.24g
2
= 0.35
m ol
4
g
3
⋅ c m
3
3
0.35 = 0.7047
si n
−1
(0.7047) = 44.8
∘
≈ cos
−1
(0.7047) = 45.19
∘
≈ 45
∘
of 34 36
This is the angle in the diagonal of a unit square (see fig. 14).
This value:
The thing to do here is to do the calculation with dimensionless values.
The units are:
Which makes sense: grams usually occupy space, grams per centimeter cubed. But here, we have not
centimeters cubed, but centimeter to the one, a straight line. Thus if you want to have grams in a straight
line (cm) you have to have the cube root of grams, because mass is three dimensional, and the line (cm) is
one dimensional. It is not out of the ordinary to present mass like this: The thin disc approximation for the
distribution of mass over a two dimensional disc can very closely approximate the mass in its actual three
dimensional manifestation, only integrating over two dimensions. That which we are seeing here is the
cross section of three dimensional atoms over a straight line, which can accurately be represented by three
dimensional atoms because atoms are so small, distributed over a line, it is as if a three dimensional atom
(a small sphere) is like a one dimensional point.
1
0.0906
0.0076g
cm
3
= 0.084
g
cm
3
0.084
g
cm
3
1
0.24
= 0.35
g
cm
3
3
0.35 = 0.7047
3
0.35 = 0.7047
3
0.35 = 0.7047
gr a m s
1/3
cm
of 35 36
Conclusion
We might say:
1. The purpose of stars is to produce not just biological elements, but artificial intelligence (AI) elements.
Because if we include AI elements in the life elements we have a mathematical pattern in the periodic
table of the elements that determines an equation that predicts the equations of nucleosynthesis in the
stars.
2. Is it the purpose of stars, as well, to produce malleable and ductile metals for humans to make tools
early on (The Iron Age) and copper wire (Age of Electronics). Or even that silicon dioxide (obsidian)
flakes into sharp edges so that we could create spearpoints with which to hunt in our beginning.
3. We are evolving towards a culminated form of specific mathematical proportions. We are evolving
towards an intermembral index of perfectly the square root of two uniformly throughout the human
species.
4. It is a purpose of biological life (C, N, O, H) to discover the properties of (P, B, Si) so it can make
computing machines which are ultimately necessary to its survival.
of 36 36
The Author
